Use calculus to find the area A of the triangle with the given vertices.

Question

(0, 0),    (3, 4),    (1, 8)

A = 

Answer 1

The answer I calculate is 10 square units.

I integrate (8x-(4/3)x)dx from x= zero to 1

I integrate (-2x+10-(4/3)x)dx from x= 1 to 3

Answer 2

The problem can be solved employing a variety of methods .

Geometric, trigonometric, using determinants up to setting and evaluating either single or double integrals to mention some.

Since the problem is listed under calculus I think using single integrals sounds reasonable.

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Let us label the three given points as O(0,0) , A(3,4) and B(1,8)

Then the lines OA, OB, and AB have equations y= (4/3 x , y= (8) x ,and y = (-2)x + 10 respectively

Then the area A =∫₀¹[8x-4x/3 ]dx +∫₁³[ (-2x + 10 − 4x/3]dx =10/3+ 20/3 =10